Math 653: Graduate Algebra       Autumn 2017

Instructor: Frank Sottile

Lectures: TuΘ 14:20–15:35 Blocker 160

Office Hours :	Mondays:	13:30–14:30
	Fridays:	10:00–11:00
	By appointment

Office: Blocker 601 K in the geometry centre

Email: sottile@math.tamu.edu Text-only with 653 in the subject line.

Book: Algebra by Hungerford. Download chapters or purchase inexpensive softcover from Springer.

Course Description This is a first semester graduate course in abstract algebra, and is intended to be an introduction to the fundamental objects of groups, rings, modules, fields, and vector spaces. I intend to cover most of Chapters I–III and parts of Chapter IV from Hungerford's classical algebra text (at right). We should cover the following topics, time permitting. basic group theory solvable groups finitely generated abelian groups Sylow theorems and basics of the classification of simple groups free groups and inverse limits Rings, integral domains, and fields commutative rings polynomial rings localization principal ideal domains and unique factorization domains power series and power series rings introduction to modules exact sequences free modules and vector spaces Prerequisites: Undergraduate abstract algebra (Math 415/6) or its equivalents.

Graduate Work:

This is the first of a two-term sequence on Graduate Algebra, which is a foundational course covering material that every mathematician should know. It forms the air one breathes in many research fields, from algebra to number theory, geometry, and algebraic geometry, and is assumed in many advanced classes in these topics. The pace will be fast and advanced for many, for there is a lot to cover. Also, as a graduate class, significantly more is expected of you than in your previous courses.

Reading:

You should read the sections of the book before they are covered in class, work over your notes after lecture (I will not review material covered in previous lectures!), and reread the sections after they are covered. Rewriting your notes is an excellent path towards full understanding (I did this in all of my graduate courses.) Some material that you are responsible for will not be covered in class, but may be found in the book and in the exercises—read those too.

You are expected to read and understand the first chapter before class starts.

Homework:

There will be weekly written homework assignments to be handed in to be graded. These wil be assigned in nearly every class; a summary may be found here. These will be marked by our grader, who is an advanced senior graduate student.

Writing Mathematics:

Developing your ability to write mathematics well is a critical skill for your future studies, and careful writing is linked to clear thinking. For example, after working out the solution to a homework problem, which is at best a very rough draft, you should then neatly write up the solution properly, omitting needless steps and falsehoods (remember, a false claim in a proof invalidates the proof), and striving for clarity and brevity. To facilitate this development, I will regularly assign and collect problems which I will mark, using a significantly more rigourous scale than perhaps that employed by our grader, which will be discussed in class. This will form part (1/4) of your grade.

Group Work:

You are encouraged to work together to find solutions to the homeworks. However, you must write up the work you turn in yourself. Also, you are absolutely forbidden to consult web sites or problem solution sources (other than the textbook and class notes). Violating this rule has serious consequences, both as it violates the universities academic integrity policy (read that here), and because not doing your own work will likely lead to an inability to do the problems on the course exams.

Exams:

The class will have two exams. I am still trying to decide on their lengths.
The mid term exam will likely be in the evening of October 5
The final exam is scheduled for December 13 from 1–3 PM in Blocker 160.

Grading:

Your course grade will be based in equal parts on regular homework, the midterm exam, homework graded by Frank, and on the final exam.